Someone is broadcasting a stream of bits. You don't know why. A 500-bit-long sample looks like this:

01100110110101011011111100001001110000100011010001101011011010000001010000001010
10100111101000101111010100100101010010101010101000010100110101010011111111010101
01010101011111110101011010101101111101010110110101010100000001101111100000111010
11100000000000001111101010110101010101001010101101010101100111001100001100110101
11111111111111111100011001011010011010101010101100000010101011101101010010110011
11111010111101110100010101010111001111010001101101010101101011000101100000101010
10011001101010101111...

The thought occurs to you to do Science to it—to ponder if there's some way you could better predict what bits are going to come next. At first you think you can't—it's just a bunch of random bits. You can't predict it, because that's what random means.

Or does it? True, if the sequence represented flips of a fair coin—every flip independently landing either 0 or 1 with exactly equal probability—then there would be no way you could predict what would come next: any continuation you could posit would be exactly as probable as any other.

But if the sequence represented flips of a biased coin—if, say, 1 came up 0.55 of the time instead of exactly 0.5—then it would be possible to predict better or worse. Your best bet for the next bit in isolation would always be 1, and you would more strongly anticipate sequences with slightly more 1s than 0s.

You count 265 1s in the sample of 500 bits. Given the hypothesis that the bits were generated by a fair coin, the number of 1s (or without loss of generality, 0s) would be given by the binomial distribution $(\frac{500}{k}) (0.5)^{k} (0.5)^{500 - k}$ , which has a standard deviation of $\sqrt{500 \cdot {0.5}^{2}} = \sqrt{125} \approx 11.18$ , so your observation of $265 - 250 = 15$ excess 1s is about $\frac{15}{11.18} \approx 1.34$ standard deviations from the mean—well within the realm of plausibility of happening by chance, although you're at least slightly suspicious that the coin behind these bits might not be quite fair.

... that is, if it's even a coin. You love talking in terms of shiny, if hypothetical, "coins" rather than stodgy old "independent and identically distributed binary-valued random variables", but looking at the sample again, you begin to further doubt whether the bits are independent of each other. You've heard that humans are biased to overestimate the frequency of alternations (10101...